4.2 Trigonometric Functions; The Unit Circle


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1 4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure. The poin P=( ) represens a poin on he uni circle. The following definiions are given based on his picure. sin csc cos sec an co If we sar wih and pu in our definiions above we will have cos sin. EXAMPLE: Suppose a poin on he uni circle is. Find all si rigonomeric values. We wan o use our definiions o answer he quesions. We know ha and because of he poin given. This auomaicall ells us ha sin and cos. We can plug in and ino he oher equaions o find he remaining rig values. csc sec an co Domain and Range of Sine and Cosine Domain is wha values we can pu ino a rig funcion () and he range is he values i reurns. sin cos Domain: Range: [ ] Range: [ ] Domain:
2 Le s look a he angle of 4 or. A his angle we end up wih he following riangle: 4 Secion 4. Noes Page Triangle 4 We can use our definiions of sine cosine and angen o find eac values: 4 sin 4 cos 4 an 4 From our above definiions we can also find he following: csc 4 sec 4 co 4 EXAMPLE: Find he eac value wihou using a calculaor: an co. 4 4 We jus need o plug in he values of he rig funcions we found above: + =. The following Even Odd Properies will allow ou o no deal wih negaive angles. Even Odd Properies cos( ) cos sec( ) sec sin( ) sin csc( ) csc an( ) an co( ) co Periodic Properies If we sar a an angle and go around one revoluion ( 60 or radians) we will end up a he same angle we sared wih. The k value is an ineger and represens how man revoluions are going around. If ou wan o use degrees replace he k in he equaions below wih 60k. sin( sin csc( csc cos( cos sec( sec an( an co( co
3 EXAMPLE: Find he EXACT value of sin4. Secion 4. Noes Page We need o divide 4 b 60 o see how man revoluions we have. If ou divide 4 b 60 ou will ge 4 wih a remainder of 4. So we can rewrie our problem as: sin( 4 60(4)). From our definiions above we know ha sin( 4 60(4)) sin 4. From our previous values we know ha sin 4. So we know ha sin 4. EXAMPLE: Find he EXACT value of 7 an. 4 Firs we will use our evenodd proper o change he negaive angle ino a posiive angle: an an 4 4. I would be easier o change his ino degrees: 76. Dividing his b 4 60 we will ge wih a remainder of 4. So our problem becomes an( 4 60()). From our periodic properies an( 4 60()) an 4. From our previous values we know ha an 4. So we know ha 7 an. 4 Fundamenal Ideniies sin csc csc sin cos sin an sec co csc cos sec sec cos sin an cos cos co sin an co co an EXAMPLE: Given sin and cos find he oher 4 rig values using ideniies. We jus use he ideniies and plug in our values for sine and cosine where necessar: csc sin / sec cos sin an cos co an
4 EXAMPLE: Given 7 sin and Secion 4. Noes Page 4 0 use he Phagorean Ideni sin cos o find cos. We firs sar wih he ideni sin cos 49 Simplifing will give ou cos 7 7 and plug in for sin. You will ge cos. 49. Now isolae he cosine b subracing from boh sides. You will ge cos. Square roo boh sides o ge cos. Since he angle for needs o be 0 we know ha his angle mus be in he firs quadran. Wha we know from his is ha he coordinae of he uni circle mus be posiive in he firs quadran so cos. EXAMPLE: The uni circle below has been divided ino equal arcs corresponding o values of: and. Use he ( ) coordinaes in he figure o find he value of each rigonomeric funcion a he indicaed real number or sae ha he epression is undefined. a.) cos The cosine value is alwas he value so we read he value from our picure above o ge cos. b.) sin The sine value is alwas he value. Since each segmen is we need o go over 4 segmens which is in he second quadran. Here we see ha sin.
5 c.) cos Secion 4. Noes Page There are wo was o solve his one. The firs mehod is o use he uni circle direcl. A negaive angle means we need o go in he clockwise direcion so we will go down ino he fourh quadran o he corresponding poin. The cosine value is alwas he value so cos. The second wa o solve his is o use he EvenOdd Proper o ge rid of he negaive. You will ge cos cos. Since he negaive is gone we can now go counerclockwise on he uni circle o he poin. The cosine value is alwas he value so cos. d.) an 6 There are wo was o solve his one. The firs mehod is o use he uni circle direcl. A negaive angle means we need o go in he clockwise direcion so we will go o he firs quadran o he corresponding poin. The angen value is alwas / so an. 6 The second wa o solve his is o use he EvenOdd Proper o ge rid of he negaive. You will ge cos cos. Since he negaive is gone we can now go counerclockwise on he uni circle o he poin. The cosine value is alwas he value so cos. EXAMPLE: Use an ideni o find he eac value wihou using a calculaor: cos sin 6 6 This can be rewrien as: cos 6. Now square everhing inside he parenhesis: subracing we ge. Therefore cos sin. 6 6 sin. Now subsiue values b using he figure on he previous page Reduce his fracion:. Afer EXAMPLE: Use a calculaor o find he approimae value of sin 4. o four decimal places. Since we are in degrees ou wan o make sure our calculaor is in degrees. I can show ou in class how o make sure ou are in he correc mode. For calculaors where ou can see wha ou are ping jus pe in his epression as i shows. Oherwise pe in he angle firs and hen he sine ke. Answer: sin
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